Indeterminate system

In mathematics, particularly in number theory, an indeterminate system has fewer equations than unknowns but an additional a set of constraints on the unknowns, such as restrictions that the values be integers.^[1]

Examples

Linear indeterminate equations

For given integers $a$ , $b$ and $n$ , a linear indeterminant equation is $ax+by=n$ with unknowns $x$ and $y$ restricted to integers. The necessary and sufficient condition for solutions is that the greatest common divisor, $(a,b)$ , is divisable by $n$ .^[1]^: 11

Conditions giving rise to indeterminacy

In linear systems, indeterminacy occurs if and only if the number of independent equations (the rank of the augmented matrix of the system) is less than the number of unknowns and is the same as the rank of the coefficient matrix. For if there are at least as many independent equations as unknowns, that will eliminate any stretches of overlap of the equations' surfaces in the geometric space of the unknowns (aside from possibly a single point), which in turn excludes the possibility of having more than one solution. On the other hand, if the rank of the augmented matrix exceeds (necessarily by one, if at all) the rank of the coefficient matrix, then the equations will jointly contradict each other, which excludes the possibility of having any solution.

Finding the solution set of an indeterminate linear system

Let the system of equations be written in matrix form as

Ax=b

where $A$ is the $m\times n$ coefficient matrix, $x$ is the $n\times 1$ vector of unknowns, and $b$ is an $m\times 1$ vector of constants. In which case, if the system is indeterminate, then the infinite solution set is the set of all $x$ vectors generated by^[2]

x=A^{+}b+[I_{n}-A^{+}A]w

where $A^{+}$ is the Moore–Penrose pseudoinverse of $A$ and $w$ is any $n\times 1$ vector.

References

^ ^a ^b Hua, Luogeng (1982). "Chapter 11. Indeterminate Equations". Introduction to Number Theory. SpringerLink Bücher. Berlin, Heidelberg: Springer Berlin Heidelberg. ISBN 978-3-642-68130-1.
^ James, M., "The generalised inverse", Mathematical Gazette 62, June 1978, 109–114.